Low-temperature Ising dynamics with random initializations

It is well known that Glauber dynamics on spin systems typically suffer exponential slowdowns at low temperatures. This is due to the emergence of multiple metastable phases in the state space, separated by narrow bottlenecks that are hard for the dynamics to cross. It is a folklore belief that if the dynamics is initialized from an appropriate random mixture of ground states, one for each phase, then convergence to the Gibbs distribution should be polynomially fast. However, such phenomena have largely evaded rigorous analysis, as most tools in the analysis of Markov chain mixing times are tailored to worst-case initializations. In this paper, we establish this conjectured behavior for the classical case of the Ising model on an $N$-vertex torus in $\mathbb Z^d$. Namely, we show that the mixing time for the Glauber dynamics, initialized in a $\frac 12$-$\frac 12$ mixture of the all-plus and all-minus configurations, is $O(N\log N)$ in all dimensions, at all temperatures below the critical one. We also give a matching lower bound, showing that this is optimal. A key innovation in our analysis is the introduction of the notion of "weak spatial mixing within a phase", an adaptation of the classical concept of weak spatial mixing. We show both that this new notion is strong enough to imply rapid mixing of the Glauber dynamics from the above random initialization, and that it holds for the Ising model at all low temperatures and in all dimensions. As byproducts of our analysis, we also prove rapid mixing of the Glauber dynamics restricted to the plus phase, and the Glauber dynamics on a box in $\mathbb Z^d$ with plus boundary conditions, when initialized from the all-plus configuration.